On the distribution of supremum of square-Gaussian random processes. (English. Russian original) Zbl 0842.60039

Kozachenko, Yu. V.; Oleshko, T. A.

Theory Probab. Math. Stat. 47, 57-64 (1993); translation from Teor. Jmovirn. Mat. Stat. 47, 55-62 (1992).

Let \(\xi (t)\in \mathbb{R}^d\), \(t\in T\), be a vector Gaussian process with zero mean and let \(A(t)\) be a symmetric nonrandom matrix. The processes of type \(\eta (t)= \xi^T (A) A\xi (t)\), as well as processes which are mean-square limits of sequences \(\eta_n (t)\) are called square-Gaussian processes. The authors prove some useful inequalities for distribution and moments of \(\sup_{t\in T} \eta(t)\). For instance, \[ P \Bigl\{\sup_{t\in T} (\eta (t)- E\eta (t))> x\Bigr\}\leq G_1 \exp \{-s_1 x\} \quad \text{for } x> x_0. \] Exact expressions and estimations for \(G_1\), \(s_1\), \(x_0\) are given.

Reviewer: N.M.Zinchenko (Kiev)

MSC:

60G15	Gaussian processes
60E15	Inequalities; stochastic orderings
46E30	Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

Keywords:

Orlicz space; pre-Gaussian process; pseudometric; epsilon-dimension; square-Gaussian process; Gaussian process; orthogonal transform